**Determinant of a Matrix**

**Determinants**
originally appeared in the study of the linear equations. We shall, however,
associate the notion of the determinant with **matrices**. For this purpose, we
have to consider the square matrices of order 1, 2, 3 etc.

If A is a
square matrix, its determinant is denoted by **|A|** or **det.(A)**. And, the value of
determinant is calculated differently for different ordered square matrices. Here
is mentioned about the calculation of determinants of square matrices of order
1, 2 and 3.

**Determinant of a 1×1 Matrix**

Let A =
(a_{11}) be a square matrix of order 1×1. Then, its determinant |A| is given
simply by the value of its single element a_{11}.

i.e. |A|
= |a_{11}| = a_{11}

For
example: Let, A = (2), then

Determinant
of A = |A| = |2| = 2

**Determinant of a 2×2 Matrix**

**Determinant of a 3×3 Matrix**

Here is
an easy pattern to find the minors of any element of a 3×3 matrix:

** Note:** Determinant of 3×3 matrix
A can also be calculated by taking the products of the elements of any row (or
column) and the corresponding cofactors. i.e.

|A| = a_{11}A_{11}
+ a_{12}A_{12} + a_{13}A_{13}

Or,

|A| = a_{21}A_{21}
+ a_{22}A_{22} + a_{23}A_{23}

Or,

|A| = a_{31}A_{31}
+ a_{32}A_{32} + a_{33}A_{33}

Or,

|A| = a_{11}A_{11}
+ a_{21}A_{21} + a_{31}A_{31}

Or,

|A| = a_{12}A_{12}
+ a_{22}A_{22} + a_{32}A_{32}

Or,

|A| = a_{13}A_{13}
+ a_{23}A_{23} + a_{33}A_{33}

All will
give the same value for the determinant of A.

Now,

|A| = a_{11}A_{11}
+ a_{12}A_{12} + a_{13}A_{13} = 1×2 + 2×1 + 0×-7
= 4

|A| = a_{21}A_{21}
+ a_{22}A_{22} + a_{23}A_{23} = 2×-4 + 3×2 +
1×6 = 4

|A| = a_{31}A_{31}
+ a_{32}A_{32} + a_{33}A_{33} = 5×2 + 4×-1 +
2×-1 = 4

|A| = a_{11}A_{11}
+ a_{21}A_{21} + a_{31}A_{31} = 1×2 + 2×-4 +
5×2 = 4

|A| = a_{12}A_{12}
+ a_{22}A_{22} + a_{32}A_{32} = 2×1 + 3×2 +
4×-1 = 4

|A| = a_{13}A_{13}
+ a_{23}A_{23} + a_{33}A_{33} = 0×-7 + 1×6 +
2×-1 = 4

Here, we
got |A| = 4 in every calculations, so we can use any one of the above methods.
Generally, we use the first method to calculate the determinant of a 3×3 matrix
A. i.e.

|A| = a_{11}A_{11}
+ a_{12}A_{12} + a_{13}A_{13}

**Mechanical Rule (Rule of Sarrus)**

**Mechanical Rule (Rule of Sarrus)**

A mechanical rule for finding the value of a third order determinant is as indicated below:-

From the
algebraic sum of the products of the elements of the diagonals pointing
downwards, subtract the algebraic sum of the products of the elements of the
diagonals pointing upwards.

This rule
of expansion of a third order determinant is known as The Rule of Sarrus.

** Note:** This rule does
not work for the determinants of the order greater than 3.

**Properties of 3×3 Determinant**

We have
seen how to evaluate a 3×3 determinant. There are some properties of 3×3
determinant. Look at them and and their verification.

**Prop. 1: The value of determinant is unaltered by interchanging
its rows and columns.**

Verification:

**Prop. 2: Interchanging any two adjacent rows (or columns)
changes the sign of the determinant.**

Verification:

**Prop. 3: If any two rows (or columns) of a determinant are identical,
then the value of the determinant is zero.**

Verification:

**Prop. 4: If all the elements of any row (or column) are
multiplied by a constant k, then the value of determinant is multiplied by k.**

Verification:

**Prop. 5: If each element of any row (or column) of a determinant
is expressed as the sum of two terms, then the determinant can be expressed as
the sum of determinants.**

Verification:

**Prop. 6: If to the elements of any row (or column) a multiple of
any other row (or column) is added, the value of the determinant remains
unaltered.**

Verification:

*Worked
Out Examples*

*Worked Out Examples*

*You can
comment your questions or problems regarding the determinants of matrices and
their properties here.*

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